Geometric Progression

A geometric progression (GP) is a progression the ratio of any term and its previous term is equal to a fixed constant. It is a special type of progression. In order to get the next term in the geometric progression, we have to multiply the current term with a fixed number known as the common ratio, every time, and if we want to find the preceding term in the progression, we just have to divide the term with the same common ratio. Example: 2, 4, 8, 16, 32, ... is a GP having a common ratio of 2.

The geometric progressions can be finite or infinite. Its common ratio can be negative or positive. Here we shall learn more about the GP formulas, and the different types of geometric progressions.

A geometric progression is a special type of progression where the successive terms bear a constant ratio known as a common ratio. It is also commonly referred to as GP. The GP is generally represented in form a, ar, ar^2.... where 'a' is the first term and 'r' is the common ratio of the progression. The common ratio can have both negative as well as positive values. To find the terms of a geometric series, we only need the first term and the constant ratio.

The geometric progression is of two types. They are

• finite geometric progression and
• infinite geometric progression.

Let us see the information about each of these.

Finite geometric progression

Finite geometric progression contains a finite number of terms. It is the progression where the last term is defined. For example 1/2,1/4,1/8,1/16,...,1/32768 is a finite geometric series where the last term is 1/32768.

Infinite geometric progression

Infinite geometric progression contains an infinite number of terms. It is the progression where the last term is not defined. For example, 3, −6, +12, −24, +... is an infinite series where the last term is not defined.

Here are the formulas related to geometric progressions. Consider a geometric progression a, ar, ar², ar³, ...

• n^th term: a_n = ar^{n - 1} (or) a_n = r a_{n - 1}
• Sum of the first n terms: S_n = a(rⁿ - 1) / (r - 1) when r ≠ 1 and S_n = na when r = 1.
• Sum of infinite terms: S_∞ = a / (1 - r) when |r| < 1 and the sum is NOT defined when |r| ≥ 1.

Let us study each formula in detail in the upcoming sections.

To find the n^th term of a GP, we require the first term and the common ratio. If the common ratio is not known, the common ratio is calculated by finding the ratio of any term to its preceding term. The formula for the n^th term of the geometric progression is:

a_n = ar^n-1

where

• a is the first term
• r is the common ratio
• n is the number of the term which we want to find.

This formula directly follows by observing the geometric progression pattern a, ar, ar², ar³, ...

The geometric progression sum formula is used to find the sum of all the terms in a geometric progression. As we read in the above section that geometric progression is of two types, finite and infinite geometric progressions, hence the sum of their terms is also calculated by different formulas.

• If the number of terms in a geometric progression is finite, then the sum of the geometric series is calculated by the formula:
  S_n = a(1 − rⁿ)/(1 − r) for r ≠ 1, and
  S_n = an for r = 1

• If the number of terms in a geometric progression is infinite, then the sum of the geometric series is calculated by the formula:
  S_∞ = a/(1 - r), when |r| < 1
  The sum cannot be found when |r| ≥ 1

Proof of Sum of Finite Geometric Progression Formula

Consider a finite geometric progression of n terms, a, ar, ar², ..., ar^{n - 1}. Then their sum is,

S_n = a + ar + ar² + ar³ + ... + ar^n-1... (1)

Multiplying both sides by r,

rS_n = ar + ar² + ar³ + ... + arⁿ... (2)

Subtracting equation (1) from equation (2),

rS_n - S_n = arⁿ - a

S_n (r - 1) = a (rⁿ - 1)

S_n = a(rⁿ - 1) / (r - 1)

Since (r - 1) is in its denominator, it is defined only when r ≠ 1. If r = 1, the progression looks like a, a, a, ... and the sum of the first n terms, in this case, S_n = a + a + a + ... (n times) = na.

Proof of Sum of Infinite Geometric Progression Formula

Consider an infinite geometric sequence a, ar, ar², ... Its sum is denoted by S_∞. Then

S_∞ = a + ar + ar² + ar³+ ... ...(1)

Multiply both sides by r,

rS_∞ = ar + ar² + ar³+ ... ... (2)

Subtracting equation (2) from equation (1),

S_∞ - rS_∞ = a

S_∞ (1 - r) = a

S_∞ = a / (1 - r)

This formula is valid only when |r| < 1. This is because when the common ratio is less than 1 (a proper fraction), the terms become smaller and smaller as we go forward and they are equivalent to 0. Hence the sum is defined in this case. But when |r| ≥ 1, then the terms become larger and larger infinitely and hence we cannot determine the sum in this case.

Geometric Progression vs Arithmetic Progression

Here are a few differences between geometric progression and arithmetic progression shown in the table below:

Important Notes on Geometric Progression:

• In a geometric progression, each successive term is obtained by multiplying the common ratio to its preceding term.
• The formula for the nth term of a geometric progression whose first term is a and common ratio is r is: a_n=ar^n-1.
• The sum of n terms in GP whose first term is a and the common ratio is r can be calculated using the formula: S_n = [a(1-rⁿ)] / (1-r).
• The sum of infinite GP formula is given as: S_n = a/(1-r) where |r|<1.

☛Related Topics:

• Geometric Series Formula
• Sum of n Terms of AP
• Geometric Progression Calculator

What are GP Formulas?

Here are the GP formulas for a geometric progression with the first term 'a' and the common ratio 'r':

• n^th term, a_n = ar^n-1.
• Sum of the first 'n' terms, S_n = a(1-rⁿ)/(1-r) when r ≠ 1. When r = 1, S_n = na.
• Sum of infinite terms (when |r| <1), S_∞ = a/(1−r). When |r| ≥ 1, we can't find the sum.

How Do You Find the Sum of an Infinite Geometric Progression?

The infinite geometric series with common ratio r such that |r| < 1 can have a sum and it can be calculated by the formula S_∞ = a/(1−r), where a is the first term and r is the common ratio. So if we want to calculate the sum of an infinite GP series, we have to use the given formula and put the value of the first term and constant ratio in the formula, and evaluate.

Which Infinite Geometric Progression has a Sum?

A geometric progression with an infinite number of terms can have two types of common ratios, first where |r| < 1, and another where |r| > 1. So the infinite geometric series with common ratio |r| < 1 has a sum equal to S = a/(1 - r) and the infinite geometric series with |r| > 1 can not have a finite sum.

What is r in GP Formula?

In geometric progression, r is the common ratio of the two consecutive terms. The common ratio can have both negative as well as positive values. In order to get the next term in the GP, we have to multiply with a fixed term known as the common ratio, every time, and if we want to find the preceding term in the progression, we just have to divide the term by the same common ratio.

How to Find the Common Ratio in Geometric Progression?

The common ratio is calculated by finding the ratio of any term by its preceding term. For example, consider the G.P.: 2, 4, 8, ... The common ratio is r = 4/2 = 2.

What is the Difference Between Arithmetic Progression and Geometric Progression?

If each successive term of a progression is less than the preceding term by a fixed number, then the progression is an arithmetic progression (AP). If each successive term of a progression is a product of the preceding term and a fixed number, then the progression is a geometric progression. The ratio of two terms in an AP is not the same throughout but in GP, it is the same throughout. To understand more differences, click here.

What is the Difference Between Geometric Progression and Harmonic Progression?

A harmonic progression (HP) is a progression obtained by taking the reciprocal of the terms of an arithmetic progression. A geometric progression (GP) is a progression where every term bears a constant ratio to its preceding term. An example of HP is 1/2, 1/4, 8, 1/16,... and an example of a GP is 2, 4, 8, 16, 32, ......

What is the Difference Between the Finite Geometric Progression and the Infinite Geometric Progression?

If there are finite terms in a geometric progression (GP), then it is a finite GP. If there are infinite terms in a GP, then it is an infinite GP. The concept of the first term and the common ratio is the same in both series.

How Do You Find the nth Term of an Infinite Geometric Progression?

The geometric progression formula is used to find the nth term in the infinite geometric progression. To find the nth term in the infinite GP, we require the first term and the common ratio. If the common ratio is not known, the common ratio can be calculated by finding the ratio of any term by its preceding term. The formula for the nth term of the GP is: a_n = ar^n-1

where

• a is the first term
• r is the common ratio
• n is the number of the term which we want to find.